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A378627 Products of 6 distinct primes that are sandwiched between semiprime numbers.

%I A378627 #24 Feb 03 2025 02:24:20
%S A378627 39270,66990,71610,79170,82110,99330,110670,122430,123690,125970,
%T A378627 129030,132090,136290,144690,152490,163590,166530,167790,180642,
%U A378627 182910,190190,191730,215670,220110,222222,226590,227766,231990,235410,239190,247170,248710,249690,254562,258258,260130
%N A378627 Products of 6 distinct primes that are sandwiched between semiprime numbers.
%C A378627 All terms are even.
%C A378627 Not all terms are divisible by 6: the first that is not is a(21) = 190190. The first term that is deficient is a(1966) = 4739702. - _Robert Israel_, Feb 03 2025
%H A378627 Robert Israel, <a href="/A378627/b378627.txt">Table of n, a(n) for n = 1..10000</a>
%e A378627 39270 is a term because 39270=2*3*5*7*11*17 is the product of six distinct primes, 39269=107*367 and 39271=173*227 are both semiprimes.
%e A378627 66990 is a term because 66990=2*3*5*7*11*29 is the product of six distinct primes, 66989=13*5153 and 66991=31*2161 are both semiprimes.
%p A378627 with(priqueue):
%p A378627 children:= proc(t) local R,i,pp;
%p A378627    R:= NULL:
%p A378627    pp:= nextprime(t[6]);
%p A378627    for i from 6 to 2 by -1 do
%p A378627      R:= R, [t[1]*pp/t[i], op(t[2..i-1]),op(t[i+1..6]),pp];
%p A378627      if t[i-1] <> prevprime(t[i]) then break fi;
%p A378627    od;
%p A378627    {R}
%p A378627 end proc:
%p A378627 Res:= NULL: count:= 0:
%p A378627 initialize(pq):
%p A378627 insert([-2*mul(ithprime(i),i=2..6),3,5,7,11,13],pq);
%p A378627 while count < 100 do
%p A378627   t:= extract(pq);
%p A378627   if numtheory:-bigomega(-t[1]-1) = 2 and numtheory:-bigomega(-t[1]+1) = 2 then
%p A378627     Res:= Res, -t[1]; count:= count+1;
%p A378627   fi;
%p A378627   for tt in children(t) do insert(tt,pq) od:
%p A378627 od:
%p A378627 Res; # _Robert Israel_, Feb 03 2025
%t A378627 SequencePosition[Array[FactorInteger[#][[;; , 2]] &, 270000] /. {2} -> {1, 1}, {{1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1}}][[;; , 1]] + 1 (* _Amiram Eldar_, Dec 02 2024 *)
%Y A378627 Intersection of A067885 and A124936.
%Y A378627 Cf. A001358, A378097.
%K A378627 nonn
%O A378627 1,1
%A A378627 _Massimo Kofler_, Dec 02 2024
%E A378627 Edited by _Robert Israel_, Feb 03 2025